Jan 22, 2013

This note provides a simple example of the typical noise-reduction available when a transimpedance (Tz) amplifier is used with a tandem post amplifier filter. The example below uses a 2nd order Butterworth filter with unity gain (at low frequency) as the post amplifier. The design f3db bandwidth of the Butterworth filter is 1.7MHz, slightly higher than the f3db bandwidth of 1.64 MHz for the transimpedance amplifier alone. The transimpedance response curves show the gain of the transimpedance amplifier alone, and with the Butterworth filter added. (The other curves in the response graph are for the transimpedance amplifier alone). The design of the transimpedance amplifier alone is approximately the Q=1/√2 case (in fact a Butterworth design itself) corresponding to maximally flat frequency response. The response of the Tz amplifier alone shows the expected 40dB/decade drop off characteristic of a 2nd order response at high frequency. WITH the Butterworth post-filter added, the final response at high frequency is 4th order with a rolloff of 80dB/decade. (Note that the combined response is not exactly a 4th order Butterworth filter response) :

The graph below compares the cumulative noise for the Tz amplifier alone and with the 2nd order post amplifier added. The significant reduction of the noise contribution due to the op-amp voltage noise source

The Butterworth post amplifier above was assumed noiseless. The achievable noise reduction will depend on the component values and op-amp used in the post-amplifier design. The results above represent the best possible noise reduction. To estimate the noise contributed by the post-amplifier, consider a single op-amp Sallen-Key implemention of a 1.7MHz 2nd order Butterworth filter is shown below:

The GBW product of the op-amp should be at least ~ 5 x f3db of the filter design bandwidth, or about 10 MHz to ensure the step filter rolloff over a sufficient high-frequency range. Since the design is a 2nd order Butterworth, it is easy to estimate the thermal noise contribution from resistor R1 of the filter circuit since it represents a flat noise voltage source at the input to the filter. For 2nd order filters with Q=0.707, the noise bandwidth is 1.111xf3db or about 1.9MHz. By comparison, the thermal noise contribution from R2 and the filter op-amp voltage noise

It is not difficult to calculate the transfer functions for noise sources at R2 and

Since a good design target with substantial phase margin is the Butterworth maximally flat response with Q=1/√2, we will assume that both the transimpedance stage alone and the LP post 2nd order filter are separately designed with this same Butterworth Q value and the same f3db frequency. For the transimpedance stage, this means selecting a proper value for the feedback capacitance Cf to target this Q value, given the other component values and the op-amp GBW. The LP filter stage could be a Sallen-Key filter stage with components chosen to obtain the same f3db and Q=1/√2 as the transimpedance stage. The overall f3db of the

For this Butterworth case, it is easy to analytically integrate the combined transfer functions to obtain the total integrated output voltage noise. These results are essentially a calculation of the total "noise bandwidth" of the combined circuit. The results for the Tz stage alone and with both stages are:

For the 2nd order Tz stage alone:

- The factor
**1.111f3db**is the**noise bandwidth**for the thermal and op-amp current noise contributions - The extra factor
**BW0/fz**for the op-amp voltage noise source, shows the strong effect of noise-gain peaking associated with input capacitance Ci and fz

- The factor
**0.833f3db**for the**noise bandwidth**of the thermal and op-amp current noise contributions is lowered to exactly 75% of the value for the Tz stage alone - The effect of
**BWo/fz**is reduced to exactly 25% of the value for the Tz stage alone, indicating the potential benefit for noise reduction

In the expressions above,